In the paper, the extended finite element method (XFEM) is combined with
a recovery procedure in the analysis of the discontinuous Poisson
problem. The model considers the weak as well as the strong
discontinuity. Computationally efficient low-order finite elements
provided good convergence are used. The combination of the XFEM with a
recovery procedure allows for optimal convergence rates in the gradient
i.e. as the same order as the primary solution. The discontinuity is
modelled independently of the finite element mesh using a
step-enrichment and level set approach. The results show improved
gradient prediction locally for the interface element and globally for
the entire domain.